Existence of multi-valued solutions with asymptotic behavior of parabolic Monge-Ampère equation
- Limei Dai1Email author
https://doi.org/10.1186/s13661-015-0307-7
© Dai; licensee Springer. 2015
Received: 20 September 2014
Accepted: 17 February 2015
Published: 4 March 2015
Abstract
In this paper, we extend the results of multi-valued solutions of elliptic Monge-Ampère equation to parabolic Monge-Ampère equation. We use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of parabolic Monge-Ampère equation. Moreover, we prove that the multi-valued solution is continuous in the whole space.
Keywords
MSC
1 Introduction
There is a vast literature on the parabolic Monge-Ampère equation (1.1); see [2, 6–11] etc. In particular, Gutiérrez and Huang [7] obtained a generalization of a theorem by Calabi. Wang and Wang [9] proved the existence of viscosity solutions by the approximation procedure and the nonlinear perturbation method, and they also obtained the regularity of the viscosity solutions.
There are many results as regards the multi-valued solutions. Evans [12] and Caffarelli [13] studied the multi-valued harmonic functions. Evans [12] has proved that the conductor potential of a surface with minimal capacity is a double-valued harmonic function. In [13], Caffarelli proved the Hölder continuity of the multi-valued harmonic functions. Jin et al. [14–16] employed a level set method for the computation of multi-valued geometric solutions to general quasilinear PDEs and multi-valued physical observables to the semiclassical limit of the Schrödinger equations. In 2006, Caffarelli and Li [17] investigated the multi-valued solutions of the Monge-Ampère equations. They first introduced the geometric situation of the multi-valued solutions and then obtained the existence, boundedness, regularity and the asymptotic behavior at infinity of the multi-valued viscosity solutions. Recently, the author and Bao [18] investigated the multi-valued solutions of the Hessian equations. The author [19] also studied the finitely valued and infinitely valued solutions to the parabolic Monge-Ampère equation \(-u_{t}\det(D^{2}u)=f\). For more detailed introduction of the multi-valued solutions and other models in nonlinear PDEs, see [17, 18] and [20]. In this paper, we extend some results of the multi-valued solutions in [17] to the parabolic Monge-Ampère equation \(-u_{t}\det(D^{2}u)=f\).
Definition 1.1
Similarly we say a function \(u\in C^{2,1}(G_{k})\) if \(D_{x}^{i}D_{t}^{j}u\) (\(i+2j\leq2\)) is continuous at \((x,t,m)\) for any \((x,t,m)\in G_{k}\).
Definition 1.2
A function \(u\in C^{0}(Q)\) is called parabolically convex in Q, if u is convex in x and nonincreasing in t.
Our purpose of this paper is to study the existence of multi-valued viscosity solutions with asymptotic behavior of the parabolic Monge-Ampère equation (1.1). We shall extend the results of the elliptic equations to the parabolic equation (1.1). The main result of this paper is the existence theorem of multi-valued solutions with prescribed asymptotic behavior at infinity.
Theorem 1.1
This paper is arranged as follows. In Section 2, we give some lemmas and in Section 3, we shall prove Theorem 1.1.
2 Preliminaries
Definition 2.1
A function \(u(x,t,m)\in C^{0}(G_{k})\) is called a viscosity solution of (2.1), if \(u(x,t,m)\in C^{0}(G_{k})\) is both a viscosity subsolution and a viscosity supersolution of (2.1).
Let \(Q=\Omega\times(0,T]\), \(\partial_{p}Q=\partial\Omega\times(0,T)\cup\overline {\Omega }\times\{0\}\) be the parabolic boundary of Q, and \(SQ=\partial\Omega\times(0,T)\) be the side boundary of Q, \(\overline{S}Q=\partial\Omega\times[0,T]\). The following lemmas and remark can be found in [19].
Lemma 2.1
Remark 2.1
If \(\bar{t}=0\), then we only need \(u=v\) on \(\overline{S}D^{\prime}\) in condition (2.2).
Lemma 2.2
Lemma 2.3
3 Existence of multi-valued solutions with asymptotic behavior
In this section, we will prove Theorem 1.1.
Proof
We divide the proof into five steps.
Step 1. We construct a viscosity subsolution of (1.2).
Step 2. We define the Perron solution of (1.2).
Step 3. We prove that \(u_{a}\) is a viscosity solution of (1.2).
Step 4. We prove that (1.3) holds.
Step 5. We prove that \(u_{a}\) satisfies (1.4).
Declarations
Acknowledgements
The author is indebted to the referees for their helpful comments. This work is supported by National Natural Science Foundation of China (11201343), Shandong Province Young and Middle-Aged Scientists Research Awards Fund (BS2011SF025).
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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